Numerical and Analytical Methods for Variational Inequalities and Related Problems with ApplicationsView this Special Issue
Research Article | Open Access
Aihong Wang, "Viscosity Approximation Methods for Equilibrium Problems, Variational Inequality Problems of Infinitely Strict Pseudocontractions in Hilbert Spaces", Journal of Applied Mathematics, vol. 2012, Article ID 150145, 20 pages, 2012. https://doi.org/10.1155/2012/150145
Viscosity Approximation Methods for Equilibrium Problems, Variational Inequality Problems of Infinitely Strict Pseudocontractions in Hilbert Spaces
We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of the solutions of the equilibrium problem and the set of fixed points of infinitely strict pseudocontractive mappings. Strong convergence theorems are established in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently.
Let be a real Hilbert space and let be a nonempty convex subset of .
A mapping of is said to be a -strict pseudocontraction if there exists a constant such that for all ; see . We denote the set of fixed points by (i.e., ).
Note that the class of strict pseudocontraction strictly includes the class of nonexpansive mappings which are mappings on such that for all . That is, is nonexpansive if and only if is a 0-strict pseudocontraction. Let be a bifunction from to , where is the set of real numbers. The equilibrium problem for is to find such that The set of solutions of (1.3) is denoted by . Given a mapping , let for all . Then the classical variational inequality problem is to find such that . We denote the solution of the variational inequality by ; that is Let be a strongly positive linear-bounded operator on if there is a constant with property A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping on a real Hilbert space : where is a linear-bounded operator, is the fixed point set of a nonexpansive mapping on , and is a given point in . The problem (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see [1–11]. In particular, Combettes and Hirstoaga  proposed several methods for solving the equilibrium problem. On the other hand, Mann , Shimoji and Takahashi  considered iterative schemes for finding a fixed point of a nonexpansive mapping. Further, Acedo and Xu  projected new iterative methods for finding a fixed point of strict pseudocontractions.
In 2006, Marino and Xu  introduced the general iterative method: for , They proved that the sequence of parameters satisfies appropriate condition and that the sequence generated by (1.7) converges strongly to the unique solution of the variational inequality . Recently, Liu  considered a general iterative method for equilibrium problems and strict pseudocontractions: where is a -strict pseudocondition mapping and , are sequences in . They proved that under certain appropriate conditions over , , and , the sequences and both converge strongly to some , which solves some variational inequality problems. Tian  proposed a new general iterative algorithm: for nonexpansive mapping with , where is a -Lipschitzian and -strong monotone operator. He obtained that the sequence generated by (1.9) converges to a point in , which is the unique solution of the variational inequality . Very recently, Wang  considered a general composite iterative method for infinite family strict pseudocontractions: for , where is a mapping defined by (2.5), is a -Lipschitzian, and -strongly monotone operator. With some appropriate condition, the sequence generated by (1.10) converges strongly to a common element of the fixed point of an infinite family of -strictly pseudocontractive mapping, which is a unique solution of the variational inequality . Kumam proposed many algorithms for the equilibrium and the fixed point problems with -strict pseudoconditions; see [14–16]. In particular, in 2011, Kumam and Jaiboon  considered a system of mixed equilibrium problems, variational inequality problems, and strict pseudocontractive mappings: where is a -strict pseudocondition mapping. They proved that under certain appropriate conditions over , , , , , , , and , the sequence converges strongly to a point which is the unique solution of the variational inequality . Inprasit  proposed a viscosity approximation methods to solving the generalized equilibrium and fixed point problems of finite family of nonexpansive mapping in Hilbert spaces.
In this paper, motivated by the above facts, we use the viscosity approximation method to find a common element of the set of solutions of the equilibrium problem and the set of fixed points of a infinite family of strict pseudocontractions.
Throughout this paper, we always write for weak convergence and for strong convergence. We need some facts and tools in a real Hilbert space which are listed as below.
Lemma 2.1. Let be a real Hilbert space. There hold the following identities:(i), (ii).
Lemma 2.2 (see ). Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that(i),
(ii) or .
Recall that given a nonempty closed convex subset of a real Hilbert space , for any , there exists a unique nearest point in , denoted by , such that for all . Such a is called the metric (or the nearest point) projection of onto . As we all know, if and only if there holds the relation:
Lemma 2.3 (see ). Let be an -Lipschitzian and -strongly monotone operator on a Hilbert space with , and . Then is a contraction with contractive coefficient and .
Lemma 2.4 (see ). Let be a -strict pseudocontraction. Define by for each . Then, as , is a nonexpansive mapping such that .
Lemma 2.5 (see ). Let be a Hilbert space and a contraction with coefficient , and an -Lipschitzian continuous operator and -strongly monotone with . Then for , That is, is strongly monotone with coefficient .
Let be a sequence of -strict pseudo-concontractions. Define . Then, by Lemma 2.4, is nonexpansive. In this paper, we consider the mapping defined by
Lemma 2.6 (see ). Let be a nonempty closed convex subset of a strictly convex Banach space , let be nonexpansive mappings of into itself such that , and let be real numbers such that , for every . Then, for any and , the limit exists.
Using Lemma 2.6, one can define the mapping of into itself as follows:
Lemma 2.7 (see ). Let be a nonempty closed convex subset of a strictly convex Banach space . Let be nonexpansive mappings of into itself such that , and let be real numbers such that , for all . If is any bounded subset of , then
Lemma 2.8 (see ). Let be a nonempty closed convex subset of a Hilbert space be a family of infinite nonexpansive mappings with , and let be real numbers such that , for every . Then .
For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions:(A1) for all ;(A2) is monotone; that is for any ;(A3)for each ;(A4) is convex and lower semicontinuous for each .
We recall some lemmas which will be needed in the rest of this paper.
Lemma 2.9 (see ). Let be a nonempty closed convex subset of , let be bifunction from to satisfying (A1)–(A4), and let and . Then there exists such that
Lemma 2.10 (see ). Let be a bifunction from into satisfying (A1)–(A4). Then, for any and , there exists such that Further, if , then the following hold:(1) is single-valued;(2) is firmly nonexpansive;(3);(4) is closed and convex.
Lemma 2.11 (see ). Let and be bounded sequences in a Banach space, and let be a sequence of real numbers such that for all . Suppose that for all . and . Then .
Lemma 2.13 (see ). Let be a Hilbert space, and let be a nonempty closed convex subset of , and a nonexpansive mapping with . If is a sequence in weakly converging to and if converges strongly to , then .
3. Main Results
Now we start and prove our main result of this paper.
Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from satisfying (A1)–(A4). Let be a family -strict pseudocontractions for some . Assume the set . Let be a contraction of into itself with , and let be a strongly positive linear bounded operator on with coefficient and . Let be an -inverse strongly monotone mapping. Let be the mapping generated by and as in (2.5). Let be a sequence generated by the following algorithm:
where , , , and are sequences in . Assume that the control sequences satisfy the following restrictions:(i) and ;(ii);
Then converges strongly to which is the unique solution of the variational inequality or equivalent , where is a metric projection mapping form onto .
Proof. Since , as , we may assume, without loss of generality, that for all . By Lemma 2.3, we know that if , then . We will assume that . Since A is a strongly positive bounded linear operator on , we have
So this shows that is positive. It follows that
Step 1. We claim that the mapping where has a unique fixed point. Let be a contraction of into itself with . Then, we have for all . Since , it follows that is a contraction of into itself. Therefore the Banach contraction mapping principle implies that there exists a unique element such that .
Step 2. We shall show that is nonexpansive. Let . Since is -inverse strongly monotone and for all , we obtain where , for all . So we have that the mapping is nonexpansive.
Step 3. We claim that is bounded.
Let ; from Lemma 2.10, we have Note that It follows that By simple induction, we have Hence is bounded. This implies that , are also bounded.
Step 4. Show that .
Observing that and , we get By Lemma 2.10, we obtain In particular, we have Summing up (3.14) and using (A2), we obtain for all . It follows that This implies It follows that Hence, we obtain where . By (3.12) and (3.19), we obtain From (2.5), we have where .
Note that where .
Suppose , then .
Hence, we have Then Combining with (i), (iii), and (iv), we have Hence, by Lemma 2.11, we obtain as . It follows that We also know that So
Step 5. We claim that .
Observe that From (A1), (A3), and (3.28), using step , we have This implies that Next we want to show .
Let ; we have Therefore Note that From (3.34), we have It follows that From conditions (i), (vi) and (3.26), we have We also compute So, from (3.39), we get It follows that So On the other hand, we also know that and hence So Hence From (i), (3.42), and (3.26), we know that From (3.37) and (3.42), we can get On the other hand, we have